Local metric homogeneity forcing periodic triangulations

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $X$ be either the Euclidean plane with its Euclidean metric or the hyperbolic plane with its standard constant-curvature metric. Let $\mathcal T$ be a locally finite geometric triangulation of $X$ whose triangles have diameter at most a fixed $r>0$. Assume that for every pair of vertices $v,w$ there is an ambient isometry of $X$ mapping $B_X(v,r)$ onto $B_X(w,r)$ and carrying the restriction of $\mathcal T$ in the first ball onto its restriction in the second. Periodicity means that a discrete group of ambient isometries preserving the entire triangulation acts cocompactly on $X$ (in the Euclidean case this gives a lattice of translation periods).

Statement recorded in the supplied source.

Must $\mathcal T$ be periodic? [S2]

Short English statement

Does identical geometric structure in all vertex-centered balls of the stated radius force a globally periodic triangulation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000062 (AMR-099-0062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Euclidean vs. Graph Metric, version dated 16 July 2012; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 5, Question 5.7 (with Romain Tessera), p. 7. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.